# Octagonal-truncated dodecahedral duoprism

Octagonal-truncated dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOtid
Coxeter diagramx8o x5x3o ()
Elements
Tera20 triangular-octagonal duoprisms, 12 octagonal-decagonal duoprisms, 8 truncated dodecahedral prisms
Cells160 triangular prisms, 30+60 octagonal prisms, 96 decagonal prisms, 8 truncated dodecahedra
Faces160 triangles, 240+480 squares, 60 octagons, 96 decagons
Edges240+480+480
Vertices480
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, (5+5)/2, (5+5)/2 (base triangle), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {45+4{\sqrt {2}}+15{\sqrt {5}}}{8}}}\approx 3.24418}$
Hypervolume${\displaystyle 5{\frac {99+99{\sqrt {2}}+47{\sqrt {5}}+47{\sqrt {10}}}{6}}\approx 410.60782}$
Diteral anglesTodip–op–odedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Tiddip–tid–tiddip: 135°
Odedip–op–odedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Todip–trip–tiddip: 90°
Odedip–dip–tiddip: 90°
Central density1
Number of external pieces40
Level of complexity30
Related polytopes
ArmyOtid
RegimentOtid
DualOctagonal-triakis icosahedral duotegum
ConjugatesOctagrammic-truncated dodecahedral duoprism, Octagonal-quasitruncated great stellated dodecahedral duoprism, Octagrammic-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(8), order 1920
ConvexYes
NatureTame

The octagonal-truncated dodecahedral duoprism or otid is a convex uniform duoprism that consists of 8 truncated dodecahedral prisms, 12 octagonal-decagonal duoprisms and 20 triangular-octagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-octagonal duoprism, and 2 octagonal-decagonal duoprisms.

## Vertex coordinates

The vertices of an octagonal-truncated dodecahedral duoprism of edge length 1 are given by all even permutations of the last three coordinates of:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right).}$

## Representations

An octagonal-truncated dodecahedral duoprism has the following Coxeter diagrams:

• x8o x5x3o (full symmetry)
• x4x x5x3o () (octagons as ditetragons)